PAPERS AND PRESENTATIONS
AN OVERVIEW OF DRAG EMBEDDED ANCHOR HOLDING CAPACITY FOR DREDGING AND
OFFSHORE APPLICATIONS
S.A. Miedema1, G.H.G. Lagers2, J. Kerkvliet3
ABSTRACT
Dredging and mining is shifting to deeper waters. For dredging with TSHD’s the limit is around 150 m of water
depth, but for mining the water depth can be hundreds or even thousands of meters. New technologies have to be
developed or copied and adapted from the offshore industry. At the Delft University, Offshore Engineering, students
carry out research into many subjects related to moorings and mooring systems, like:
1.
2.
3.
4.
5.
6.
7.
8.
9.
The holding capacity of drag anchors in sand.
The holding capacity of drag anchors in clay.
The holding capacity and operations of suction anchors.
The use of the catenary equation in moorings.
The methodology about choosing the right anchor for different purposes.
The methodology about choosing the right anchor line for different purposes and conditions.
The methodology about designing an anchoring system for FPSO’s.
The methodology about designing an anchoring system for SPAR’s.
The methodology about designing an anchoring system for Semi-Sub’s.
The results of the research can also be found on: http://www.offshoreengineering.org. In dredging and offshore
anchors are used for positioning, but also for operations (the cutter suction dredger). In all cases it is evitable that a
proper estimation of the holding capacity is very useful when designing the application.
The holding capacity of anchors depends on the digging depth, the soil mechanical properties end of course the
dimensions and the shape of the anchor. The digging depth also depends on the soil mechanical properties and the
shape of the anchor. Now the first question is of course how is the digging depth related to the soil mechanical
properties and the shape of the anchor and the second question is, how does this relate to the holding capacity.
By means of deriving the equilibrium equations of motion of the anchor and applying the cutting theories, the
digging behavior of anchors can be simulated. The main challenges are how to model the shape of the anchor and
how to apply the existing cutting theories to this complex shape. This paper gives a first attempt to derive
equilibrium equations based on the cutting theory of Miedema (1987).
Keywords: Dredging, anchors, holding capacity, soil mechanics
INTRODUCTION
This paper is the result of assignments carried out by students for the Offshore Moorings course of the MSc program
Offshore Engineering of the Delft University. In this study an analyses is made of the penetration behavior of an
anchor in sand. The following points are to be taken into account.
• The geometry of the anchor and how to simplify it.
• What happens when the anchor penetrates the soil?
• Which forces will occur during penetration?
• How to solve this mathematically?
First of all, the most common anchors on the market are analyzed and a general anchor geometry will be chosen.
This chosen geometry will be simplified to a 2D geometry, which will be realistic for a first analysis. After this step
the penetration behavior of an anchor will be described in different phases, such a way that it is clear and easy to
1
Associate Professor, Delft University of Technology, s.a.miedema@3me.tudelft.nl
Associate Professor, Delft University of Technology, g.h.g.lagers@tudelft.nl
3
Student in Offshore Engineering, Delft University of Technology. JohnKerkvliet@wanadoo.nl
2
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PAPERS AND PRESENTATIONS
understand. Forces on the soil layer, fluke, shank and mooring line forces will be defined and analyzed. The forces
will be described as a function of the geometries including the relevant variable angles.
THE GEOMETRY OF THE ANCHOR
While searching for the best simplified anchor geometry, knowledge of the most common anchors on the market is
needed. Vrijhof Anchors 2005 gives a good overview of the most common anchors. Two types of anchors can be
considered, horizontal load anchors and vertical load anchors. The vertical load anchor can withstand both
horizontal and vertical mooring forces. The horizontal anchor or drag embedment anchor can only resist the
horizontal loads. The drag embedment anchor is mainly used for catenary moorings, where the mooring line arrives
the seabed horizontally. The vertical load anchor is used in taut leg mooring systems, where the mooring line arrives
at a certain angle the seabed. A good starting-point for this case is to analyze the horizontal load anchor, because this
anchor is often used and will form an adequate challenge. To determine a simplified geometry of this horizontal
anchor an actual figure of this anchor is needed. In the Figures 1 and 2, a sketch of the selected anchor can be found.
Figure 1. Examples of anchors.
Figure 2. The geometry of anchors.
The actual blade (fluke) on the anchor that will penetrate the soil is represented by the horizontal part of the above
given figure. The shank is represented by the other part of the anchor. On the end of the anchor an anchor shackle
can be found. A first simplification will be made by modeling the anchor as a 2D model. Hereby all the calculations
will be made easier, but the geometry is still to complex to determine all the forces. A second simplification can be
made by supposing the anchor as two straight lines as can be seen Figure 3. This simplification is allowed, because
this is a conceptual (first) model. When all the forces and the penetration curve of this simple model are known, a
much more complicated model can be made. Figure 4 gives an overview of drag embedded anchors.
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PAPERS AND PRESENTATIONS
Figure 3. The definition of fluke and shank.
For the rest of the case a few assumptions must be made. First, the type of soil will be sand, this way the cohesion
and adhesion effects can be neglected. As a next assumption, it is considered that the anchor penetrates the soil at a
very low velocity. Therefore inertia and water tension can also be neglected. Several constrains were also made to
simplify the 3D force analysis into a 2D analysis. This way several shear zones can be neglected. As a final
assumption, the force acting on the point of the fluke will be neglected. This force is low considering a big anchor
and will be fully cancelled by the force perpendicular on the shank.
Figure 4. Typical drag embedded anchors (API 2005).
SOIL RESISTANCE TO EMBEDDED ANCHOR LINE
The forces on the embedded anchor line can be seen as a separate system in the anchor burial process and will be
treated first. Up until 1989 the work reported on embedded anchor chains was basically theoretical. Values of the
design parameters such as the effective chain width in sliding (Bs), the effective chain width in bearing (Bb) and the
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PAPERS AND PRESENTATIONS
bearing capacity factor in clay (Nc) were suggested, but little experimental proof confirmed their validity. With
respect to general practice, the effective chain widths can be expressed in terms of the nominal chain diameter (D):
B s = EWS ⋅ D
(1)
(2)
B b = EWB ⋅ D
where EWS and EWB are the parameters to express the effective widths in sliding and bearing, respectively.
Vivatrat et al. (1982) developed the analytical model of a chain inside soil by assuming the chain length inside soil
as a summation of short line segments and expressing the equilibrium conditions of each segment. The effective
width parameters EWS and EWB proposed were 10 and 2.6, respectively and the use of a value for Nc between 9
and 11 was suggested.
Yen and Tofani (1984) performed laboratory measurements on sliding and bearing soil resistances on a ¾ inch
diameter stud-link chain in very soft silt. During cutting and sliding tests, the maximum soil resistances were
established to be mobilized within a small movement of chain, even less than half a link. The EWS parameter can be
determined from laboratory tests and varies between 5.7 and 8.9. The Nc factor at a particular depth was found to be
between 7.1 and 12.1, considering the EWB parameter to be 2.37.
Using the finite segment approach, Dutta (1986, 1988) derived the nodal equilibrium equations for chain segments
and proposed a simple calculation method. This study showed good agreement with the results obtained by the
analytical method used by Vivatrat et al. (1982).
Degenkamp and Dutta (1989) derived a more accurate analytical model of embedded chain under soil resistance and
a simple calculation procedure. They used a soil model to accurately predict the soil resistances to the chain inside
soil and estimated critical design parameters, such as effective widths of chain, based on laboratory tests.
Assuming: (1) Chain elements are inextensible; (2) due to the chain shift, the soil medium suffers an undrained
loading condition; (3) soil in the vicinity of the chain reaches limit state of stress and thereby develops ultimate soil
resistances; and (4) the shear strength and weight of the soil over a chain element are constant. The forces on a chain
element are as presented in Figure 5.
Figure 5. Force system on an embedded anchor line.
Assuming Δs is small, equilibrium in tangential direction leads to:
∂T
= −(f + w sin Φ 1 )
∂s
(3)
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PAPERS AND PRESENTATIONS
and by using the incremental integration approach, one can write:
T2 = T1 − Δs(f + w sin Φ 1 )
(4)
Equilibrium in normal direction ( Δs is small) leads to:
∂φ P − w cos Φ 1
=
∂T
∂s
Δs
T+
∂s
(5)
or, for discrete elements, applying the incremental approach:
Φ 2 = Φ1 +
(p − w cos Φ 1 )Δs
T2
(6)
The entire embedded chain configuration can be assumed to be a summation of the discrete chain elements. For each
element, the value of T2 and Φ 2 can be determined using the known values of T1 , Φ 1 , w , f and p . To
determine f and p , the tangential movement has been assumed to cause an uncoupled sliding resistance,
independent of the normal soil resistances. Using this assumption, the frictional resistance is written as:
f = B s αS u
(7)
And the normal soil resistance is written as:
p = Bbq
(8)
The value of q is evaluated using the formula given by Skempton (1951) for the ultimate net soil resistance of a
strip footing:
⎛
h ⎞
⎟ (with a maximum N c = 7.6 )
q = N c S u Where N c = 5.14⎜⎜ 1 + 0.2
B b ⎟⎠
⎝
(9)
From these equations, it can be seen that the accuracy of p and f are governed by the factors (EWB x N c ) and
(EWS x α ) respectively. Extensive testing led to the next values:
• For very soft (Su = 5 kPa) clay:
EWB = 2.5,
EWS = 8.0
• For firm (Su = 34 kPa) clay:
EWB = 2.3,
EWS = 7.2
Grote (1993) used the exact same approach and values obtained by Degenkamp and Dutta (1989) to determine the
force distribution and geometric profile of the embedded anchor line in his work to simulate the kinematic behavior
of work anchors.
Neubecker and Randolph (1994, 1995) derived closed form expressions for both the load development and chain
profile to avoid the numerical solution by an incremental integration technique used by Degenkamp and Dutta (1989)
and simplify the procedure. Their work corroborates the results found by Degenkamp and Dutta. For the typical case
where the chain angle at the seabed is zero, the expression becomes:
Ta θ a2
= DQ
2
(10)
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PAPERS AND PRESENTATIONS
The expression for frictional development along the chain was derived as:
T0
= e μθa
Ta
(11)
In addition they found formulations for the embedded anchor line in sand using the same approach, but changing the
bearing capacity factor. For non-cohesive soils, the bearing pressure q may be expressed in terms of a standard
bearing-capacity factor N q as:
q = N q γ 'z
(12)
THE KINEMATIC BEHAVIOR OF DRAG ANCHORS IN CLAY
Grote (1993) derived a dynamic model to describe the anchor embedment. He described the forces on the anchor as
illustrated in Figure 6. The forces on the anchor consist of its weight, shear and normal forces on both fluke and
shank and the tension force from the anchor line pulling the anchor. The fluke force ff is calculated with a
transformation of the cutting formula of Miedema 1987. Grote states that the force on the fluke depends on the depth
and velocity of the anchor and a constant, containing soil specific properties. This leads to a quantitative formula of
this force:
ff = C(depth )a ( velocity )b
(13)
Figure 6. Forces on an embedded anchor.
The second force on the fluke, fff , is a front force on the fluke caused by the bearing capacity of the surrounding
soil. This same force is present at the shank of the anchor and called fsf . They are both calculated with the formula
for the bearing resistance of a strip footing formulated by Terzaghi:
fff = fsf = A front (cN c +
γB
N γ + γdN q )
2
(14)
For no free-draining soils as clay, it can be assumed that the internal friction angle φ = 0 and N q = 1 . The second
term is small compared to the last one, so equation 16 can be simplified to:
fff = fsf = A front (cN c + γd )
(15)
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PAPERS AND PRESENTATIONS
fss is a shear force on the shank. In clay ground the angle of internal friction is assumed zero while the cohesion is
not. The shear stress where failure occurs is then equal to the cohesion, which is equal to the undrained shear
strength.
fss = A shear s u
(16)
And finally, ft is the force on the pad eye implied by the anchor line.
Steward (1992) published methods to describe the kinematic behavior of drag anchors in cohesive soils. These
methods were simplified by Neubecker and Randolph (1995) who formulated bearing capacity and moment
equilibrium calculations utilizing two fundamental anchor resistance parameters, f and θ w . The assumption that a
drag anchor travels parallel to its flukes is widely accepted. Therefore the authors expressed a geotechnical
resistance force Tp acting on the anchor parallel to the direction of the fluke, illustrated in Figure 7, as:
Tp = fA p N c s u
(17)
Since there will also be geotechnical forces normal to the fluke, the resultant resistance force Tw acting on the
anchor will make an angle θ w with the fluke.
Figure 7: anchor parameter, θ w
Therefore:
Tw =
Tp
(18)
cos θ w
The pad eye force Ta can be determined as the resultant of Tw and the anchor weight Wa . As the anchor embeds,
the upper fluke surface and resultant force Ta will be at angles of β and θ a respectively to the horizontal.
Allowing for the offset angle ψ between Tw and Ta , it then follows that:
β = θ w + ψ − θa
(19)
Neubecker and Randolph (1994, 1995) also developed an expression for the anchor chain tension and angle at the
anchor padeye assuming the chain angle at the seabed is zero:
Ta θ a2
= DQ
2
(20)
Every anchor can be considered to have unique properties f and θ w that are independent of the anchor size or the
soil strength profile, and can be determined by experimental modeling or comparison against published field data.
The equations described above can then be implemented into an incremental simulation as follows:
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PAPERS AND PRESENTATIONS
1.
2.
3.
4.
5.
6.
Assume an anchor fluke orientation β and displace the padeye horizontally an increment Δx .
Calculate the new embedment depth D, assuming motion parallel to the previous fluke orientation.
Calculate the anchor resistance Ta from the anchor characteristics, anchor orientation and local soil strength.
Calculate the chain angle θ a using equation 20.
Calculate the new fluke angle β using equation 19 in order to maintain equilibrium.
Displace the padeye a further increment Δx and loop to step 2.
By adjusting the variables involved, the authors showed agreement with centrifuge and full-scale tests.
Thorne 1998 developed a theory from geotechnical principals, without the use of any site or anchor specific
correlations. His predictions of the anchor movement showed good agreement with nine full scale tests covering
three different sites and five anchor types. The equations used are based on the proposition that no movement will
occur until the soil forces acting parallel to the fluke are overcome. The motion of the anchor results in the soil
around the shank failing in bearing capacity on the underside and in shearing on the base and sides, exerting on the
shank the maximum force of which the soil is capable. This is also true for other elements which have to be dragged
through the soil like shackles, palms and stabilizers. These forces are calculated as:
Drag = DA i DFi S u
(21)
where DA i and DFi are the area and drag factor for the “ith” component, acting at an angle α i to the plane of the
fluke on the anchor element (Figure 8).
Figure 8. The anchor model.
Figure 9. The force application points.
There are three drag components: base bearing and skin adhesion of the base (acting at right angles to the element),
and skin adhesion on the sides (acting parallel to the fluke). The drag factors are shape dependant and taken from
geotechnical research on soil forces on cylinders, flat plates, wedges and strip footings. All forces are assumed to act
at the centers of the respective areas and the undrained shear strength is taken as that at the centre of area. The forces
of each element can be added to give the total drag force components normal and parallel to the fluke and the
moments about the fluke centre of area:
i=n
TDFN = ∑ − DA i DFi S u sin α i
(22)
i =1
i =n
TDFN = DA f DFf S u + ∑ DA i DFi S u cos α i
(23)
i =1
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PAPERS AND PRESENTATIONS
i=n
TDFM = ∑ DA i DFi S u ( Distx i sin α i + Disty i cos α i )
i =1
(24)
Considering an anchor with its fluke at an angle θ to the horizontal, the centre of area of the fluke at a depth D
below the seabed and the anchor chain at an angle θ a to the horizontal (Figure 9), the equilibrium equations are:
.
Ta cos(θ + θ a ) = TDFP − W sin θ
(25)
Fn = Ta sin( θ + θ a ) − W cos θ − TDFN
(26)
M = Ta S x sin( θ + θ a ) − S y cos( θ + θ a ) + TDFM − W( Yw sin θ + X w cos θ)
(27)
{
}
To solve this force system with the four unknowns M , Ta , Fn , θ a , one more equation is needed. Thorne used the
closed form expression given by Neubecker and Randolph 1995 for the anchor shackle tension, Ta , and the angle of
the chain at the anchor to the horizontal, θ a :
Ta θ a2
= DQ
2
(28)
The proposed approach for progressive penetration is based on the pressure distribution over a flat plate. An anchor
fluke is considered, B long and L wide, at some position acted upon by a normal force, Fn , and a moment, M ,
which results in the idealized contact pressure distribution abcd and stress changes as the plate is moved a distance
δ s (Figure 10).
Figure 10. The pressure distribution.
These stress changes can be considered as a loading on the plate consisting of an equivalent normal force and
moment acting at the centre of area.
⎧ ⎛
f δ ⎞ 2f δ (B − δ s ) ⎫
Feq = L ⎨δ s ⎜ f n + f m − m s ⎟ − m s
⎬
B ⎠
B
⎭
⎩ ⎝
(29)
⎧⎪ δ (B − δ ) ⎛
f δ ⎞ f δ 2 (B − δ s ) ⎫⎪
s
M eq = L⎨ s
⎜ fn + fm − m s ⎟ − m s
⎬
2
B ⎠
B
⎝
⎪⎩
⎪⎭
(30)
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PAPERS AND PRESENTATIONS
Now the incremental normal and angular deflections of the fluke after the movement δ s are assumed to be
proportional to Feq and M eq with proportionality constants K 1 and K 2 respectively. K 1 and K 2 are functions
of the elastic modulus of the soil, the plate size and the relative depth of the plate. If the plate moves a total distance
S
S parallel to its plane, there will be
increment and the resulting total deflections of the centre of area, δ n and
δs
δ θ are:
S
Feq
δn = K1
δs
δs → 0
S
6M
= K 1LSB(f n − f m ) = K 1 (Fn −
)
B
B
S
M eq
δθ = K 2
δs
δs → 0
K 2 LSB(f n + f m )
S⎛F B
⎞
=
= K 2 ⎜ n + 3M ⎟
2
B⎝ 2
⎠
(31)
(32)
To allow estimation of the response of a real anchor fluke, an equivalent rectangular fluke is used with the same
centre of area and absolute first moment of area. For a rectangle, the values of K 1 and K 2 are based on research
done by Lee (1962), Whitman and Rickart (1967), Butterfield and Bannerjee (1971) and Rowe and Davis (1982) on
deflection coefficients for a buried plate (Figure 11):
Figure 11. The deflection coefficients for a buried plate.
⎛ 0.6B + 0.3L ⎞
⎟
K 1 = I d ⎜⎜
⎟
⎝ BLE u Ec ⎠
K2 =
(33)
ImId
(34)
Eu Ec B 2 L
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PAPERS AND PRESENTATIONS
The term I m takes account of the aspect ratio of the rectangle, I d takes account of the depth effect, E c is a
reduction factor to take account for post yield behavior and E u is the undrained elastic modulus of the soil.
The analysis of progressive movement now proceeds as follows. Assume the fluke moving a distance S , parallel to
its plane, from an old position (n-1) to a new position n and rotating an angle δ 'θ from the initial angle θ n −1 to an
angle θ n . The statics can be solved to find Fn and M for equilibrium. Now using the progressive penetration
approach as described above, it is possible to estimate the angular displacement δ θ which would occur if this
combination of Fn and M were applied to the plate. Repeating this for various angular displacements until
δ θ = δ 'θ will result in the angle θ for this step. The drag distance (horizontal movement), IDD , of the anchor from
position n-1 to n is then calculated as:
IDD =
S sin θ n −1
+ δ n sin θ n
⎛ θ n −1 + θ n ⎞
tan⎜
⎟
2
⎝
⎠
(35)
and the new depth for step n+1, Dn +1 , is calculated as below and the whole process is repeated.
Dn +1 = Dn − δ n cos θ n + S sin θ n
(36)
A value of S = 1.0B was adopted for calculation.
Finally the American Petroleum Institute 2005 gives a graph for the holding capacity in soft clay based on the ratio
of the weight of the anchor and the holding capacity. This graph is shown in Figure 12, and gives lines for different
types of anchors. Of course this graph shows a worst case scenario based on soft clay.
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PAPERS AND PRESENTATIONS
Figure 12. Anchor holding capacity in soft clay (API 2005).
THE KINEMATIC BEHAVIOR OF DRAG ANCHORS IN SAND
Compared to the behavior of an anchor in clay, the weight of the soil above the anchor plays an important role in
non cohesive soils like sand.
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PAPERS AND PRESENTATIONS
Le Lievre and Tabatabaee 1981 proposed a limit equilibrium method. This method has been shown to give
reasonable predictions of the ultimate holding capacity of drag anchors in sand, for a given depth of embedment.
However, several assumptions in the analytical procedure make it unsuitable for application to a drag anchor during
embedment, and hence the approach does not allow prediction of the depth to which the anchor will embed, and thus
the actual capacity. A schematic representation of the failure mode and force system adopted by Le Lievre and
Tabatabaee is shown in Figure 13.
Figure 13. Soil failure mode and force system.
The proposed solution procedure followed in a stepwise manner:
1.
2.
3.
Assume a failure wedge angle, λ , of the soil.
For the particular failure wedge angle, λ , calculate the mobilized mass of soil, Ws .
For the particular failure wedge angle, λ , calculate the side friction, SF. The side friction is assumed to be
the force required to overcome the lateral earth force acting on the soil wedge area, hence:
SF =
∫ K s γz tan(φ)dA
(36)
Area
4.
5.
6.
From the force polygon of the soil wedge plus anchor (Figure 13), solve for the two unknowns of chain
tension, Ta , and soil reaction, R .
From either of the other two force polygons of soil wedge only, solve for the values of shank force, Fs , and
fluke force, Ff .
Go back to step 1 and repeat with a different value of λ . Stop procedure after minimum chain tension is
calculated. This value is the best upper bound solution to the problem.
Grote (1993) defined a force model shown in Figure 10. His problem approach is the same for clay as for sand
grounds only the formula’s for the ground resistances are different.
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PAPERS AND PRESENTATIONS
Figure 14. Forces on an embedded anchor.
The fluke force ff is calculated with a quantitative formula:
ff = C(depth )a ( velocity )b
(37)
The second force on the fluke, fff , is a front force on the fluke caused by the bearing capacity of the surrounding
soil. This same force is present at the shank of the anchor and called fsf . They are both calculated with the formula
for the bearing resistance of a strip footing formulated by Terzaghi:
fff = fsf = A front (cN c +
γB
N γ + γdN q )
2
(38)
where N c , N γ and N q are the bearing capacity factors, c the cohesion of the soil, γ the unit weight of the soil, B
the width of the footing (here fluke and shank) and d the depth of the bottom of the footing. For free-draining soils
such as sand, the cohesion c is thought to be zero. The second term is small compared to the last one, so equation
38 can be simplified to:
fff = fsf = A front ( γdN q )
(39)
To take account for the under pressure that can occur during anchor burial, the author used a empirical equation
which is a combination of the foundation theory of Terzaghi and the cutting theory of Miedema set up by Becker et
al. 1992:
fff = fsf = A front N q ( γ soil
d
+ γ water Δp )
2
(40)
where Δp is the pore under pressure that follows from the cutting theory of Miedema.
fss is a shear force on the shank. In sand ground the horizontal stress is assumed to be a function of the vertical
ground force:
σ = Kγ soil d
(41)
Using this relation, the formula used for the shear force is:
fss = A shear Kγ soil d tan( φ steel − soil )
(42)
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PAPERS AND PRESENTATIONS
For K the value 2 was taken. This value applies for piles driven into sand that densifies at the pile tip due to the
driving vibrations. And finally, ft is the force on the pad eye implied by the anchor line.
Neubecker and Randolph 1995 based their approach on the method of LeLievre and Tabatabaee 1981. They
extended this method to incorporate:
• a more realistic 3-dimensional failure pattern in the soil,
• a force acting on the back of the fluke.
The latter modification is particularly important at shallow penetrations, when the bearing capacity of the anchor
shank is insufficient to provide equilibrium (Figure 15).
Figure 15: General force model
The authors suggested that the force on the shank is dependent on its size and shape and should be calculated from a
bearing capacity viewpoint. Thus:
Fs = A s γd s N qs
(43)
1
The normal self-weight term ( bγN γ ) is omitted in equation 44, because it is assumed to have a relatively small
2
contribution and even the N qs alone tends to over predict the shank resistance.
The 3-dimensional failure mode
Dickin 1988 presented an overview of some of the various methods that have been developed to evaluate the pullout
resistance of a flat plate (Figure 16).
Figure 16. Slip surfaces used for flat plate pullout: (a) Majer 1955; (b) Vermeer and Sutjiadi 1985.
It was considered that the simple method of Majer (1955) consistently underestimated the pullout capacity of the flat
plate, while the model of Vermeer and Sutjiadi (1985) gave predictions that compared well with observations.
Neubecker and Randolph incorporated the model of Vermeer and Sutjiadi into the drag anchor problem as illustrated
in Figure 17.
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PAPERS AND PRESENTATIONS
Figure 17. 3-dimensional failure mode.
The fluke can be thought of as being mapped onto the soil surface by the displacement vector of the soil wedge. The
failure planes are inclined at the dilatation angle, ψ , to the displacement vector so that, when they reach the soil
surface, the distance they are away from the fluke shadow is proportional to the depth of the original point. This 3dimensional soil failure mode is still an idealized failure mode for the soil. However it does result in a more realistic
description of the failure surface. The area, A , can be written as:
A=
H 2 − h 2 H 2 tan λ
+
2 tan β
2
(44)
A simplification is made in the calculation of the cross-sectional area, A , in that a vertical slip surface is assumed
behind the fluke, rather than an inclined one. This is supposed to have a minor effect on the balance of forces, as low
active pressures are involved. However, no shear force is assumed across this surface and as such the extra soil mass
included by this idealized failure mode counteracts the shear force that is neglected from the realistic failure mode.
The lateral extent of the failure wedge, X , is determined from simple geometry.
X=
H tan ψ
cos(λ − ψ )
(45)
Using a pyramidical approximation for the sides of the wedge, the mobilized soil mass, Ws , is expressed as:
2 ⎞
⎛
Ws = A⎜ γB + γX ⎟
3 ⎠
⎝
(46)
The side friction, SF , that is to be used in the limit equilibrium equations is obtained from Vermeer and Sutjiadi
1985.
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PAPERS AND PRESENTATIONS
SF =
γL( H + h ) 2 (sin φ' − sin ψ )
(47)
4 cos ψ(1 − sin φ' sin ψ )
Figure 18 and 19 show the system of forces with free body diagrams and force polygons for the soil wedge only, the
anchor only and the combined anchor-soil body.
Figure 18. Free body diagram.
Figure 19. Force polygons.
The limit equilibrium calculation procedure is executed in the same way as that of Le Lievre and Tabatabaee 1981 in
that the chain tension is calculated for a given failure wedge angle, λ , which is than varied until the chain tension
reaches a minimum. The procedure begins however, by initially examining force equilibrium on the soil wedge only.
The soil mass, Ws , and the side friction, SF , are calculated using equations 46 and 47, respectively, for the 3dimensional soil wedge. The shank force, Fs , is calculated from the standard bearing capacity calculations from
equation 43. The two unknown forces on the soil wedge, which are the fluke force, Ff , and the soil reaction, R ,
can be calculated from horizontal and vertical force equilibrium requirements. The force equilibrium of the anchor
only is considered now. The fluke force, Ff , is defined from the previous equilibrium calculation, the weight of the
anchor is known and the shank force, Fs , is calculated from equation 43. Again, there are only two unknown forces
left for solution by force equilibrium, namely the force on the back of the fluke, Ffb , and the chain tension, Ta . The
failure wedge angle, λ , is then varied and the process repeated to obtain a minimum upper bound estimate of Ta .
The solution of this force system still proceeds using simple limit equilibrium methods, however, the equilibrium
solution is applied in a two step manner to arrive at a chain load and the introduction of an extra unknown and an
extra equation has resulted in a more realistic force system acting on the anchor.
Finally the American Petroleum Institute 2005 gives a graph for the holding capacity in sand based on the ratio of
the weight of the anchor and the holding capacity. This graph is shown in Figure 20, and gives lines for different
types of anchors.
261
PAPERS AND PRESENTATIONS
Figure 20. Anchor holding capacity in sand (API 2005).
Miedema et al 2006 derived a model based on 4 different penetration phases. The model is described below.
262
PAPERS AND PRESENTATIONS
PENETRATION PHASES
Phase 1: No penetration
In this first situation the anchor lies on the bed of soil and the fluke/shank angle will be considered as a minimum.
When pulling on the mooring line the anchor will scratch over the seabed, see Figure 21. A bed of soil will be
formed in front of the fluke and will give some resistance. Because of this resistance, an angle κ will reach its
maximum at a certain point. At that certain point, the bed of soil in front of the fluke will give his highest resistance
and it will become easier to penetrate than scratching over the seabed. When the assumption of a perfect sharp fluke
point is made, the point load can be neglected.
Figure 21. The anchor on top of the soil in phase 1.
Phase 2: Penetration causes fluke forces
When the fluke starts to penetrate (Figures 22 and 23), the cutting theory of Miedema 1987, can be used. Forces that
will play a role in the force balance are the fluke forces. When the angles on the fluke are considered, a few
assumptions can be made. First of all the fluke/shank angle κ will be constant and will have its maximum value. The
internal friction and the external friction angles are also constant. These parameters are only depending on the
material of the anchor and soil mechanical properties.
Figure 22. The fluke penetrating the soil in phase 2.
Fluke
Shear zone in the soil
L
x
Figure 23. The fluke in the soil.
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PAPERS AND PRESENTATIONS
G
N2
+
S2
δ
K2
S1
α
K1
β
φ N1
Figure 24. The forces on the soil layer.
As discussed before the cohesion, adhesion, inertial forces and water tension can be neglected. According to Figures
24 and 28, a force balance can be calculated.
The shear force and the normal force are related according:
S1 = N1 tan ϕ
S2 = N 2 tan δ
(48)
(49)
The grain forces will be:
(S + N )
K = (S + N )
K1 =
2
1
2
2
2
1
2
(50)
2
(51)
2
The weight of the soil can be given as a force according:
⎛ x 2 sin 2 α x 2 sin 2 α ⎞
G=⎜
+
⎟ ⋅γ
2 tan β ⎠
⎝ 2 tan α
(52)
Horizontal equilibrium of forces:
K1 ⋅ sin ( β + ϕ ) − K 2 ⋅ sin (α + δ ) = 0
(53)
Vertical equilibrium of forces:
− K1 ⋅ cos ( β + ϕ ) + G − K 2 ⋅ cos (α + δ ) = 0
(54)
T
Fluke
K2
S2
δ
+
N2
α
Fp
Figure 25. The forces on the fluke.
264
PAPERS AND PRESENTATIONS
The force K2 on the fluke is important to determine the horizontal and vertical acting forces on the fluke (Figure 25).
K2 =
G ⋅ sin ( β + ϕ )
sin (α + β + ϕ + δ )
(55)
The following forces are acting on the fluke blade:
• The Horizontal Force
Fh = K 2 ⋅ sin (α + δ )
•
(56)
The Vertical Force
Fv = K 2 ⋅ cos (α + δ )
(57)
The force Fp can be neglected as discussed before.
Phase 3: Penetration causes fluke and shank forces
In this situation the fluke is completely covered by sand and the shank will become an extra factor which will cause
penetration resistance, see Figures 26 and 27. For the shank the cutting theory of Miedema 1987, can’t be used. The
strip footing theory, as described in Verruijt (2000), will be used for determining the shank resistance. The
maximum shank resistance acts when the complete shank is penetrated.
Figure 26. The shank penetrating the soil in phase 3.
In this paragraph you will find the modeling of the forces on the fluke and the shank. The influences of the angles
will be given. The cutting theory of Miedema (1987) is still valid for the fluke part of the anchor forces. For
determining the forces on the shank, the strip footing theory, as described in Verruijt (2000), will be used. For phase
3, two shear zones are taken into account. This will lead to a geometry as shown in Figure 28.
Shank
y
Fluke
L
= Shear zone in the sand
Figure 27. The fluke and shank in the soil.
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PAPERS AND PRESENTATIONS
Figure 28. Forces on the soil layer.
As discussed before; the cohesion, adhesion, inertial forces and water tension can be neglected. For the Figures 28
and 30, a force balance can be calculated. The shear force and the normal force are related according:
The shear force and the normal force are related according:
S1 = N1 tan ϕ
S2 = N 2 tan δ
(58)
(59)
The grain forces will be:
(S + N )
K = (S + N )
K1 =
2
1
2
2
2
1
2
(60)
2
(61)
2
The weight of the soil can be determined by using the geometry of Figure 29, so the weight of the soil will be:
G = A ¦¦¦¦¦ ⋅ γ
(62)
Figure 29. The dimensions of the soil layer.
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PAPERS AND PRESENTATIONS
with:
⎛
y*sin(κ -α )+L*sin(α ) ⎞ ⎛ y 2 *sin(κ -α
A ¦¦¦¦¦ := (y*sin(κ -α )+L*sin(α ))* ⎜ L*cos(α )+
⎟−⎜
tan(β )
⎝
⎠ ⎝ 2*tan(β )
⎛ ( y*sin(κ -α )+L*sin(α ) )2 ⎞
1 2
− *L *sin(α )*cos(α ) − ⎜
⎟
⎜
⎟
β
2
2*tan(
)
⎝
⎠
(63)
Forces on the fluke and the shank
For the determination of the forces on the fluke and the shank, Figure 30, two different theories will be used. For the
fluke the cutting theory of Miedema is valid, therefore forces on the soil layer and on the fluke are the same as
discussed in phase 2. For the forces on the shank the strip footing theory can be used. This theory is based on the
fundamentals of Brinch Hansen and is a generalization of the Prantl theory. This can be found in Verruijt 2000.
P
T
+
Shank
Fluke
K2
S2
δ
N2
α
Fp
Figure 30. The forces on the anchor.
To determine the friction Brinch Hansen force P on the shank we can make use of:
P = ic sc cN c + iq sq qN q + iγ sγ
1
γ BNγ
2
(64)
Where c is cohesion and q is the external load on the soil
Because c and q are zero in this case (no cohesion and no external force on the soil), P will only be a function of the
soil weight part of the function, so:
P = iγ sγ
1
γ BNγ
2
(65)
i
Hereby γ is a correction factor for inclination factors of the load. The factor
the load.
i
sγ is a shape factor for the shape of
In this case, only a load perpendicular to the soil will be considered, so γ will be removed from the formula.
Inserting
Nγ :
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PAPERS AND PRESENTATIONS
⎛
⎛ 1 + sin φ π tan φ ⎞
B⎞
P = ⎜1 − 0,3 ⎟ B 2 yγ ⎜
e
− 1⎟ tan φ
y⎠
⎝ 1 − sin φ
⎠
⎝
(66)
Now the friction part of the shank has to be determined. For the friction of the shank, the next formula is valid:
Ffriction = σn tan(δ) y h
(67)
It is possible now to plot the results for P and Ffriction (see Figure 31) then it is possible to find out if the downward
force of the fluke is big enough to pull the shank through the seabed and further.
It is also possible now to make a total force and moment balance, to predict the trajectory of the anchor.
T
P+Fr
ε
Shank
Fluke
Fh
Fv
Figure 31. The forces on the anchor.
Phase 4: Penetration causes fluke forces, shank forces and mooring line forces
Figure 32. The mooring line penetrating the soil in phase 4.
The fluke and the shank are completely covered by sand (Figures 32 and 33). When there is still no equilibrium, a
part of the mooring line will enter the soil. The mooring line penetration will lead to an extra factor which will cause
penetration resistance. The anchor becomes stable when there is a balance between the vertical and horizontal forces
on the anchor part, which is covered by sand.
In this phase you will find the modeling of the forces on the fluke, the shank and the mooring line. The cutting
theory of Miedema is still valid for the fluke part of the anchor forces. For determining the forces on the shank and
the mooring line we will use the strip footing theory as discussed in Verruijt.
268
PAPERS AND PRESENTATIONS
m
y
Shank
Fluke
= Shear zone in the sand
L
Figure 33. Fluke, shank and mooring line in the soil.
The soil layer properties can be interpreted in a same way as described in phase 3. So this way the function for G is
still valid.
M
C
P
+
K2
S2
δ
N2
α
Fp
Figure 34. Forces on the fluke, shank and mooring line.
To determine the forces on the anchor for this situation the theory as discussed in phase 3 is valid. For the mooring
line forces (Figures 34 and 35) we will also use the Brinch Hansen theory as discussed in Verruijt 2000.
The penetration of the mooring line causes resistance perpendicular to this line (penetration resistance, see Figure
36). This effect is noticeable in all soil conditions. The type of mooring line will determine the value of this
resistance. Think of a wire rope mooring line which penetrates deeper (less resistance) than a chain mooring line.
During the penetration process of the anchor, the resistance increases when depth increases, which is related to the
position of the anchor.
The mooring line penetration can be described by the following geometry: When looking at the point where the
anchor becomes stable, a force and moment balance can be made out of all the forces on the anchor and mooring
line. In fact this is the moment were the anchor reaches his maximum holding capacity.
269
PAPERS AND PRESENTATIONS
Figure 35. The forces on the anchor.
Figure 36. The forces on the mooring line.
Vertical equilibrium of forces:
Fv − Pv − Frv − M v − Tv = 0
(68)
Horizontal equilibrium of forces:
Fh + Ph + Frh + M h − Th = 0
(69)
Moment balance to point A:
1
1
1
1
− Fv ⋅ L cos α − Fh ⋅ L sin α + ( Pv + Frv ) ⋅ y cos (κ − α ) + ( Ph + Frh ) ⋅ y sin (κ − α )
2
2
2
2
+Tv ⋅ y cos (κ − α ) + Th ⋅ sin (κ − α ) = 0
270
(70)
PAPERS AND PRESENTATIONS
CONCLUSIONS AND RECOMMENDATIONS
In the penetration behavior the different forces and moments in all four phases can be described with the theory dealt
with, in this document. These forces and moments are a function of the anchor geometry. The holding capacity of
the anchor is described as well as a function of the depth and the geometry. To predict the trajectory of the anchor
during the penetration, one has to find a relationship between the different forces and moments on the anchor and
the trajectory of the anchor. The anchor trajectory will stop when the different forces are in equilibrium, or when the
pull force will be too high for that particular anchor, at a certain depth. In the last case, the pull force necessary to
penetrate deeper in the soil, is higher then the maximum holding capacity of that particular anchor at a certain depth.
to that point the maximum holding capacity is reached (see Figure 37). When pulling further, the anchor will be
pulled out and looses his function.
Figure 37. Holding capacity vs. anchor trajectory.
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PAPERS AND PRESENTATIONS
LIST OF SYMBOLS USED
As
B
Bs
Bb
c
C
d
D
f
f
h
h
L
M
Nc
Nγ
Nq
p
q
Q
Su
Δs
T
Ta
T0
T1
T2
w
y
z
α
β
ϕ
δ
χ
γ
κ
φ
μ
α
σn
Φ1
Φ2
θa
λ
The area of the shank
Width of the shank
The effective chain width in sliding
The width of the footing, which is the chain width in bearing
Cohesion
Catenary force
Depth of bottom footing
the pad eye embedment depth
Anchor form factor
soil frictional resistance per unit chain length
Height of the shank
the depth of the footing.
Total fluke length
Resistance on mooring line
Baring-capacity factor for clay
Bearing cpacity factor
Bearing capacity factor
soil normal/bearing resistance per unit chain length
the normal ultimate soil pressure
the average bearing resistance of the chain in the soil
the undrained shear strength of the soil
chain element length
Anchor pull force
the chain load at the anchor pad eye
the chain load at the seabed
chain tension at the top of the chain element
chain tension at the bottom of the chain element
effective chain weight in soil per unit chain length
Length of the shank in the sand
Depth
Angle of the fluke in the sand
Angle of the shear zone, inclination fluke
Internal friction angle of the sand
External friction angle fluke/sand
Length of the fluke in the sand
Density of the in situ sand
Angle between fluke and shank
Internal friction angle sand
the friction coefficient between the chain and soil
a reduction factor (= 1 for soft clay)
Normal stress on the area of the shank
chain angle at the top of the chain element
chain angle at the bottom of the chain element
the chain angle at the anchor padeye
Failure wedge angle
272
m2
m
m
m
kPa
kN
m
m
kN/m
m
m
m
kN/m
kN/m
kPa
kN
kPa
m
kN
kN
kN
kN
kN
kg/m
m
m
rad
rad
rad
rad
m
ton/m3
rad
rad
N/m2
rad
rad
rad
rad
PAPERS AND PRESENTATIONS
REFERENCES
API (American Petroleum Insitute) 2005, “Design and analysis of stationkeeping systems for floating
structures”. Recommended practice 2SK 2005.
Degenkamp, G. and Dutta, A., 1989. “Behaviour of Embedded Mooring Chains in Clay During Chain
Tensioning.” Offshore Technology Conference, Paper 6031.
Degenkamp, G. and Dutta, A., 1989. “Soil Resistances to Embedded Anchor Chain in Soft Clay.” Journal of
Geotechnical Engineering Vol 115, No. 10
Det Norske Veritas., 2002. “Design and Installation of Plate Anchors in Clay.” Recommended Practice DNVRP-302.
Dunnavant, T.W. and Kwan, C-T.T., 1993. “Centrifuge Modeling and Parametric Analysis of Drag Anchor
Behavior.” Offshore Technology Conference, Paper OTC 7202.
Grote, B.J.H., 1993. “Simulation of Kinematic Behavior of Workanchors.” Faculty of Mechanical Engineering,
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University of Technology.
House, A., 1998. “Drag Anchor and Chain Performance in Stratified Soils.” Geomechanics Group, Report 1319,
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Lammes, R.R. and Siemers, R.W., 1988. “Soil Mechanics of Drag Embedment Anchor in Sand.” Faculty of
Civil Engineering, Offshore Engineering Major, Delft University of Technology.
Lin, K. and Randolph, M.F., 1998. “Numerical Analysis of Stresses and Displacements in Layered Soil
Systems.” Geomechanics Group, Report G1334, The University of Western Australia.
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Delft University of Technology, September 15th 1987.
Miedema, S.A. , Kerkvliet, J., Strijbis, D., Jonkman, B., Hatert, M. v/d, 2006, "THE DIGGING AND
HOLDING CAPACITY OF ANCHORS". WEDA XXVI AND TAMU 38, San Diego, California, June
25-28.
Mierlo, R. v., 2005. “Anchor Trajectory Modeling.” Faculty of Mechanical Engineering, Offshore Engineering
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Murff, J.D., Randolph, M.F. & Elkhatib, S., Kolk, H.J., Ruinen, R.M., Strom, P.J. and Thorne, C.P., 2005.
“Vertically Loaded Plate Anchors for Deepwater Applications.” Frontiers in Offshore Geotechnics
ISFOG.
Ruinen, R.M., 2005. “Influence of Anchor Geometry and Soil Properties on Numerical Modeling of Drag
Anchor Behavior in Soft Clay.” Frontiers in Offshore Geotechnics ISFOG.
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Geotechnical Journal 33: 574-583
Neubecker, S.R. and Randolph, M.F., 1996. “The Kinematic Behavior of Drag Anchors in Sand.” Canadian
Geotechnical Journal 33: 584-594
Neubecker, S.R. and Randolph, M.F., 1995. “Performance of Embedded Anchor Chains and Consequences for
Anchor Design.” Offshore Technology Conference, Paper OTC 7712.
Neubecker, S.R. and Randolph, M.F., 1995. “Profile and Frictional Capacity of Embedded Anchor Chains.”
Journal of Geotechnical Engineering Vol. 121, No. 11
Neubecker, S.R. and Randolph, M.F., 1995. “The Performance of Drag Anchor and Chain Systems in Cohesive
Soil.” Geomechanics Group, Report G1168, The University of Western Australia.
Neubecker, S.R. and Randolph, M.F., 1994. “Profile and Frictional Capacity of Embedded Anchor Chains.”
Geomechanics Group, Report G1142, The University of Western Australia.
O’Neill, M.P. and Randolph, M.F., 1998. “The Behaviour of Drag Anchors in Layered Calcareous Soils.”
Geomechanics Group, Report G1329, The University of Western Australia.
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Geomechanics Group, Report G1346, The University of Western Australia.
Stewart, W.P., 1992. “Drag Embedment Anchor Performance Prediction in Soft Soils.” Offshore Technology
Conference, Paper OTC 6970.
Thorne, C.P., 1998. “Penetration and Load Capacity of Marine Drag Anchors in Soft Clay.”
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274